Europhys. Lett., 74(6), pp. 1053–1059 (2006) DOI:10.1209/epl/i2005-10591-2
Kondo lattice without Nozi` eres exhaustion effect
K. Kikoin1 andM. N. Kiselev2
1 Ben-Gurion University of the Negev - Beer-Sheva 84105, Israel
2 Institute for Theoretical Physics, University of W¨urzburg D-97074 W¨urzburg, Germany
received 17 November 2005; accepted in finalform 13 April2006 published online 12 May 2006
PACS.71.27.+a – Strongly correlated electron systems; heavy fermions.
PACS.75.10.Pq – Spin chain models.
PACS.75.30.Mb – Valence fluctuation, Kondo lattice, and heavy-fermion phenomena.
Abstract. – We discuss the properties of layered Anderson/Kondo lattices with metallic electrons confined in 2Dxyplanes and local spins in insulating layers forming chains in thez direction. Each spin in this model possesses its own 2D Kondo cloud, so that the Nozi`eres’ ex- haustion problem does not occur. The high-temperature perturbational description is matched to exact low-T Bethe-ansatz solution. The excitation spectrum of the model is gapless both in charge and spin sectors. The disordered phases and possible experimental realizations of the model are briefly discussed.
The famous exhaustion problem formulated by Nozi`eres [1] states that the number of electrons eligible to participate in Kondo screening is not enough to screen magnetic moments localized in each site of periodic Anderson lattice (AL) or Kondo lattice (KL). In spite of his latest revision [2] based on mean-field 1/N expansion, this exhaustion is a stumbling stone on the way from exactly solvable Anderson or Kondo impurity model [3] to the 3D AL/KL models, which are believed to be the generic models for heavy-fermion materials [4]. The problem arises already for concentrated Kondo alloys, where the number or localized spinsNi
is comparable with the number of sitesN=Ln in the n-dimensionallattice. In this case the number of spin degrees of freedom provided by conduction electrons in a KL is not enough for screeningNilocalized spins. As an option a scenario of dynamical screening was proposed [5,6], where only part of spins screened by Kondo clouds form magnetically inert singlets. The low- temperature state of such KL is a quantum liquid, whereNssinglets are mixed withN−Ns
“bachelor” spins, which hop around and exchange with singlets thereby behaving as effective fermions. Nozi`eres’ exhaustion is measured by a parameter pN = Ni/(ρ0TK) (the number of spins per screening electron). HereTK ∼ρ−10 exp[−1/(ρ0J)] is the energy scale of Kondo effect,J is the exchange coupling constant in the single-impurity Kondo Hamiltonian,ρ0 is the density of states on the Fermi level of metallic reservoir.
Second obstacle, which does not allow the extrapolation of Kondo impurity scenario to KL is the indirect RKKY exchangeIjj between the localized spins, which arises in the 2nd order inJ or in the 4th order inV (hybridization parameter in the generic AL Hamiltonian).
The corresponding energy scale is
I=J2χcjj ∼ρ0J2, (1)
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where χcjj =N−1
qχc(q) exp[iq·Rjj] andχc(q) is the spin susceptibility of the electron gas. The Fourier transform ofχc(q) is an oscillating function, which strongly depends on the distanceRjj. IfI <0 at an average inter-impurity distance, and|I| ∼TK, then the trend to inter-site antiferromagnetic coupling competes with the trend to the one-site Kondo singlet formation (Doniach’s dichotomy [7]).
This competition prompted several possibilities to bypass the exhaustion limitations. Ac- cording to a scenario offered in [8,9], in the criticalregion|I| ∼TKof Doniach’s phase diagram, where the magnetic correlations are nearly suppressed by the on-site Kondo coupling, the spin liquid phase enters the game. This phase is characterized by the energy scale
I(T) =J2χsjj(T), (2) where χsjj is the spinon susceptibility and I(T) is renormalized due to Kondo processes exchange integral. The condition I(TK)>TK is easily achieved both in the 3D and 2D case.
The Kondo screening is then quenched in the weak-coupling regime at T > TK, so that the spin degrees of freedom remain decoupled from the electron Fermi-liquid excitations both at high temperatures T TK and at low temperatures T TK (Curie and Pauli limit for magnetic response, respectively). At T→0 the KL behaves as a two-component Fermi liquid with strongly interacting charged electrons and neutral spinons [10]. This scenario develops on the background of strong AF correlation. It includes the possibility of ordered magnetic phases with nearly screened magnetic moments and, in particular, the quantum phase transitions. Due to separation of spin and electron degrees of freedom, Luttinger’s theorem in its conventional Fermi-liquid form is invalid in this state: f-electrons represented by their spin degrees of freedom give no contribution in the formation of the electron Fermi surface. Such state is referred as a “small Fermi surface regime” in the current literature.
Another scenario for small Fermi surface regime was proposed in [11]. This scenario appeals to systems where the magnetic order is either fragile or entirely absent due to magnetic frustrations (e.g., to triangular lattices). A spinon gap carrying unit flux of Z2 gauge field is expected to arise in spin subsystem, and this gap prevents formation of Kondo singlets for a finite range of TK. As a result, Nozi`eres’ exhaustion does not occur, and fractionalization of excitations into spin-fermions and electrons exists as in the previous case. A possibility of forming the spin liquid with U(1) gauge group and spin density wave ground state has also been pointed out in [11].
In the present paper, we propose an alternative paradigm for fermion fractionalization in Kondo lattices, which possesses the generic properties of KL but is not subject to the ex- haustion limitations. Namely, we consider the structures, where the number of reservoirs for Kondo screening is the same as the number of spins in a KL. This paradigm may be realized in strongly anisotropic Kondo lattices, where the metallic electrons are confined in 2D planes interlaid by insulating layers containing magnetic ions. Then the 3D reservoir of screening electrons is defragmented intoLplanar reservoirs. Each plane still possesses the macroscopic number of spin degrees of freedom ∼L2 enough for Kondo screening, provided the concen- tration of magnetic centers per metallic plane remains small. The spin liquid features may be observed in these systems if the distribution of magnetic centers is also anisotropic, namely, if they form chains oriented in thezdirection, and the inter-chain interaction is negligibly small.
We postpone the discussion of experimental realization of such systems to the concluding section and begin with the theoreticaldescription of an idealconfiguration, where allchains penetrate the stack in zdirection (fig. 1a). The AL Hamiltonian for the quasiperiodic model of conduction electrons confined in metallic layers (xyplane), and magnetic ions localized in
(a)
spins j j+1
l l+1
layers
(b)
Fig. 1 – (a) Layered lattice of spatially separated charges in planes and spins in chains. (b) A fragment of a chain with Kondo clouds formed as “shadows” in metallic layers.
insulating layers between metallic planes is
H =
lkσ
kc†lkσclkσ+
jσ
dndjσ+1
2U ndjσndj¯σ
+
+
jl
kσ
Vkc†l+1kσ(djσ+dj+1,σ) + H.c.
. (3)
Here k is a 2D wave vector, the discrete indices numerate metallic layers l with a lattice constanta and magnetic sites j along the chains with a spacing az.The coupling constant Vk characterizes hybridization between itinerant 2D electrons in a planel+ 1 and localized states in two adjacent sites j, and j + 1 of the chain (fig. 1b). We treat the electrons in metallic planes in terms of Bloch waves clkσ, while the localized electrons are characterized by Wannier functionsdjσ. The periodicity of magnetic sites in thexyplane is not demanded, but the average distanceλbetween the impurities within a layer exceeds the radius of Kondo cloud,i.e. satisfies the conditionλvF/TK (vF andTK are Fermi velocity of 2D electrons and Kondo temperature, respectively). There is no interaction between the chains under this condition, and a single chain represents thez component of excitation spectrum. Besides, all chains contribute to thexy-component of the spin and charge response of the AL. The effects associated with the inter-chain exchange will be discussed in the concluding part.
We came to a situation where L two-dimensionalFermi reservoirs, each with capacity L2, screen Ni magnetic moments arranged in such a way that the effective concentration of these moments per metallic layer isni =Ni/L2 and satisfies the condition nia2 1. This capacity is enough to form a screening Kondo cloud for each magnetic site within a given layerl+ 1 independently of all other sites belonging to the same layer. On the other hand, two magnetic ions localized one above another in neighboring insulating layersj,j+ 1 share the same metallic screen (see fig. 1b). Replicating these dimers along thez-axis, one comes to a system of spin chains, interacting with a system of metallic layers stacked up in thexy plane. Elimination of the hybridization termV in the Hamiltonian (3) in accordance with the standard Schrieffer-Wolff procedure, results in effective exchange Hamiltonian for each chain,
Hintcd =
Ni
m=1,k,k
Jkk!skkmS!m (4)
(see fig. 1b with 2m→j, 2m−1→l). Here we use notations: S!m=12d†2m,σ !σσσd2m,σ,!skkm =
12[c†2m−1,kσ+c†2m+1,kσ]!σσσ[c2m−1,kσ+c2m+1,kσ]. The exchange integralis Jkk∼Vk∗Vk/U. We assumed the periodic boundary conditions, namelyS!N =S!1 and!sN =!s1, which imply
equal number of spins and planes. The open and twisted boundary conditions may be also imposed. Subtle effects modifying the ground state in these cases will be considered elsewhere.
Thus, the originalmodelis reduced to the anisotropic KL formed by the system of 1D spin chains penetrating the stack of 2D metallic layers. Each spin creates two Kondo clouds in adjacent planes, and two neighboring spins see each other through a metallic screen by means of indirect RKKY-like exchange. This exchange may be either ferromagnetic (FM) or antiferromagnetic (AFM). The latter case is considered in terms of Doniach’s dichotomy [7].
In our model this dichotomy should be reformulated. Since the long-range AFM ordering is impossible in 1D chain, two competing phases are Kondo singlet and spin liquid. Complete Kondo screening is not forbidden by Nozi`eres’ exhaustion principle, since the 2D screening layer is available for each spin in the chain. The Kondo screening is characterized by the energy scale TK. Thus, the competing phases in the anisotropic KL are the Kondo singlet phase and the homogeneous spin liquid of RVB type with the energy scale given by eq. (2).
In order to describe the Doniach-like phase diagram we adopt the method of [9]. Namely, we derive an effective action functionalby integrating out all“fast” fermionic degrees of free- dom with the energies∼D0, where 2D0is the conduction bandwidth. The “slow” modes give us a hydrodynamic action. Due to strong quasi-1D anisotropy there is no need in appealing to the mean-field approximation. After elimination of conduction electrons with the energies D0> ε > T in metallic layers, the couplingJ is enhanced,J →J˜= 1/(ρ0ln(T /TK)) and the indirect RKKY-like spin-spin interaction mediated by the in-plane electrons [12] arises along the chains:
Hintdd =−I
j,σσ
d†jσdj+1,σd†j+1,σdjσ. (5) HereI is defined in eq. (1) withχcj,j+1(R) =N−1
qχc(q) expi(qR),Rcharacterizes the relative distance between two spin projections on a plane. If R= 0, the chains are straight and the interaction between spins is ferromagnetic. A modelof single spin chain penetrating the stack of 2D metallic planes may be mapped on that of FM spin chain interacting with an array of 1D metallic wires [13]. The Kondo screening develops similarly to a two-site Kondo model [14]. Behavior of dilute system of FM coupled spin chains interacting with arrays of 1D fermions deviates from the two-site Kondo scenario. It will be discussed elsewhere. For R ∼ a the chains have a zigzag shape, with AFM interaction. Leaving the FM case for further studies, we concentrate here on the array of AFM coupled chains. SinceV /U 1, we adopt the nearest-neighbor approximation for RKKY interaction.
Up to this moment we treated the spin chains in a single-site approximation. This approx- imation is legitimate untilT TK ∼I. To move further, we decouple the Euclidean action of the model(4), (5),
A= β
0 dτ
j
¯cG0−1c+ ¯dD0−1d
−Hintcd −Hintdd
(6)
by means of the Hubbard-Stratonovich scheme [15] in terms of the fields [9, 16]
∆j,j±1→
σ
d†jσdj±1,σ+ c.c.
, φl→
kσ
c†l−1,kσ(dj,σ+dj+1,σ) + c.c.
. HereG0−1=∂τ−(−i∇) +µandD−1loc=∂τ−iπ/(2β) are bare inverse single-particle Green’s functions (GF) for conduction electrons and local spins, respectively,β = 1/T. The fieldφde- scribes the single-site Kondo screening and the field ∆ stands for the spinon propagation along
∆−
Πsl2 ∆+
(a)
φ− ΠK2 φ+
(b)
∆+φ−
∆− φ+
(c) Π
4K
Fig. 2 – Loop expansion for non-local action (7). Solid and dashed lines in Π2 and Π4 stand for electron and spinon propagators, respectively.
the 1D spin chain with AFM coupling. The single occupancy constraintd†j↑dj↑+d†j↓dj↓=1 is preserved at each site in the chain by the semi-fermionic transformation [17]. These two fields resolve Doniach’s dichotomy, because the long-range AFM order is absent in 1D.
We appeal to the uniform resonance valence bond (RVB) spin liquid state [16] and treat the spinon modes as fluctuations around the homogeneous solution in ann- approximation,
∆j,j±1→= ∆uj,j±1−∆ with ¯¯ ∆2(β)=β−1β
0∆(τ)∆(−τ)dτ. For this sake, we add and subtract
∆ in the inverse GF. The non-l ocal inverse spinon GF¯ D−1=∂τ−∆j,j±1−iπT /2 has to be expanded in terms of ∆−∆. Now the two interacting components of bose-like modes in two-¯ sublattice chain are spinons and Kondo clouds represented in effective action by ∆j,j+1∆j+1,j and φl,l+mφl+m,l (m = 0,1), respectively. The charged φ-mode acquires dispersion due to the non-locality of ˜Jlj, while the in-plane dispersion of conduction electrons in Kondo clouds is integrated out. The neutralspinon mode is dispersive by its origin. Then we come to an effective action with separated charge and spin sectors:
Aef f =
jl,ωn
|φl,l(ωn)|2 J˜lj
+|∆j,j+1(ωn)|2 I
+ (7)
+Tr log(G0−1) + Tr log
D−1(∆j,j+1) +G0φ∗l,lφl,l±1+ c.c. .
The last term in (7) may be represented as a loop expansion. The two first diagrams are shown in fig. 2. To calculate the diagrams, we use the non-local spinon GFDj,j+r0 (ωn) = (Dloc−1−∆)¯ −1 with cosine-like dispersion
D0j,j+r(ωn) = exp
−|r|
ln
∆¯ ωn−√
ω2n+ ¯∆2
−iπ2 i
ω2n+ ¯∆2 .
Here r numerates sites in the chain, ωn = 2πT(n+ 1/4) on the imaginary axis [17]. D0 is characterized by a branch cut at [−∆¯,∆]. In the limit ¯¯ ∆ πT it rapidly falls down with growing |r| as Dj,j+r0 (ωn) ∼ ∆¯|r|/(iωn)|r|+1. Thus the main contribution comes from Dj,j0 (ωn) = 1/i
ωn2+ ¯∆2 andDj,j±10 (ωn) = (ωn/
ωn2+ ¯∆2−1)/∆.¯
The polynomial effective action after the loop expansion acquires the form Aef f =
jjωn
|∆jj(ωn)|2
I −Πsl2|∆jj(ωn)−∆|¯ 2
+
+
jjl,ωn
1 J˜jl
−ΠK2 +ΠK4 |∆jj(ωn)|2
|φlj(ωn)|2+Tr log(G0−1)+Tr log[(D0)−1]+O(|φ|4). (8)
(a) (b)
Fig. 3 – (a) Disordered anisotropic Kondo lattices. (b) Formation of spin ladder from interacting chains.
The polarization loops Π2 and Π4 are shown in fig. 2. The action (8) is gauge invariant in accordance with Elitzur theorem [16], and spin and charge modes are separated both in the 3D lattice and in the Fock space.
To estimate ¯∆, we refer to the properties of spin chains with AFM coupling [18]. The quasi–
long-range order in these chains may be treated in terms of boson excitations in Luttinger liquid (LL) or fermion pairs in spin liquid. The spin susceptibility of a chain,∆+∆−ω=0∼∆¯2, acquires Pauli form atT∗∼8J2/EF [19], so we assume ¯∆∼T∗ in our estimates. This means that even in the criticalregion of Doniach’s diagram,TK≈J˜2/EF,the spins are “molten” into spin liquid atT ∼TK, and there is no crossover to a strong Kondo coupling regime at lowT. Evaluation of Π2,Π4in the limitπT ∆ gives Π¯ K2 ∼ρ0ln( ¯∆/T) and ΠK4 ∼ρ0/∆¯2. This leads to reduction of indirect exchange, ˜I=I[1+I/( ¯∆ l n( ¯∆/TK))]−1.The main manifestation of weak Kondo screening in the LL limit at T→0 is the reduction of LL sound velocity, v = ˜Iaz. As to the in-plane charge excitations, the formation of Kondo clouds is quenched at T TK, so instead of coherent Fermi liquid regime,φ+φ−ω→0 behaves as a relaxation mode∼[−iω/Γ +αq2+ l n( ¯∆/TK)]−1,where Γ, αare numericalconstants.
These features of two-component electron/spin liquid manifest themselves in thermody- namics. The logarithmic corrections ∼ln−1(T∗/T) are expected in low-T Pauli-like suscepti- bility of isotropic spin chains, whereas the log-corrections to the susceptibility of charged layers are quenched as ln( ¯∆/TK). The overdamped relaxation mode should be seen as a quasielastic peak inχ0. The 1D spinons contribute to the linear-T term in specific heat thus mimicking the heavy-fermion behavior, while the contribution of Kondo clouds is frozen at lowT.
In real anisotropic crystals one may expect formation of distorted and dangling chains (fig. 3a) instead of an “ideal” lattice (fig. 1). Distortion means shift of two neighboring Kondo
“shadows” in a stack. This effect may be modelled by a random overlap factorwj in RKKY integrals,Ij=wjI. The dangling bond effect meanswj= 0. Bond disorder may be treated in terms of random AFM chains [20]. According to this theory, the disorder results in transforma- tion of singlet RVB liquid into a random-singlet RVB phase with arbitrarily long singlet bonds.
More interesting effects associated with quantum criticality appear for S = 3/2 [21] where disorder-driven transition occurs between two different random singlet realizations. In the systems with integer spins of magnetic atoms the interplay between Haldane gap formation, effects of disorder and underscreened Kondo effect takes place. In chains with broken bonds the gaps arise due to the finite-length effect, so the short chain segments do not contribute to the low-T thermodynamics. With increasing impurity concentration, the Kondo clouds begin to overlap and two-leg ladders with diagonal bonds arise along with isolated chains (fig. 3b).
Nozi`eres’ exhaustion is stillnot actualfor these clusters. With further increase of the concen- tration of magnetic sites, Doniach’s problem restores in its full glory. In case of FM coupling I, true long-range order emerges in spin chains, but Nozi`eres’ exhaustion is still quenched.
One may point out the class of layered conducting/magnetic hybrid molecular solids, where the possible candidates for the application of the above theory should be looked for. These crystals are formed by alternating metallic cationic layers and insulating magnetic anionic layers with various molecular groups as building blocks containing transition metal ions as carriers of localized spins [22]. Organic cations with magnetic ions in such systems form ordered stacks. The problem is in finding systems with metallic layers where the Kondo screening lengthvF/TK is less than the distance between magnetic ions. It is worth noting, however, that the crystals containing dicyanamide radicals with Mn ions may form planar Kagome sublattice [22], thus being a promising object for the realization of the fractionalized Fermi liquid scenario proposed in [11].
∗ ∗ ∗
We are gratefulto A. Abrikosov, N. Andrei, A. Finkel’stein, L. Glazman, A. Millis, J. Schlueter andA. Tsvelik for usefuldiscussions. This work is supported by SFB-410 project, ISF grant, A. Einstein Minerva Center and the TransnationalAccess program # RITA-CT-2003-506095 at the Weizmann Institute of Sciences. MK acknowledges support through the Heisenberg program of the DFG and is gratefulto Argonne National Laboratory for the hospitality during his visit. Research in Argonne was supported by US DOE, Office of Science, under Contract No. W-31-109-ENG-39.
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